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| {{DISPLAYTITLE:Dynamics And Logic}} | | {{DISPLAYTITLE:Dynamics And Logic}} |
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| + | '''Note.''' Many problems with the sucky MathJax on this page. The parser apparently reads 4 tildes inside math brackets the way it would in the external wiki environment, in other words, as signature tags. [[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 18:00, 5 December 2014 (UTC) |
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| ==Note 1== | | ==Note 1== |
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| & \quad & | | & \quad & |
| \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q | | \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q |
− | \end{matrix}</math> | + | \end{matrix}\!</math> |
| |} | | |} |
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| <math>\begin{array}{rcc} | | <math>\begin{array}{rcc} |
| \operatorname{E}X & = & X \times \operatorname{d}X | | \operatorname{E}X & = & X \times \operatorname{d}X |
− | \end{array}</math> | + | \end{array}\!</math> |
| |} | | |} |
| | | |
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| & = & | | & = & |
| \{ \texttt{(} \operatorname{d}q \texttt{)},~ \operatorname{d}q \} | | \{ \texttt{(} \operatorname{d}q \texttt{)},~ \operatorname{d}q \} |
− | \end{array}</math> | + | \end{array}\!</math> |
| |} | | |} |
| | | |
− | The interpretations of these new symbols can be diverse, but the easiest | + | The interpretations of these new symbols can be diverse, but the easiest option for now is just to say that <math>\operatorname{d}p\!</math> means "change <math>p\!</math>" and <math>\operatorname{d}q</math> means "change <math>q\!</math>". |
− | option for now is just to say that <math>\operatorname{d}p</math> means "change <math>p\!</math>" and <math>\operatorname{d}q</math> means "change <math>q\!</math>". | |
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| Drawing a venn diagram for the differential extension <math>\operatorname{E}X = X \times \operatorname{d}X</math> requires four logical dimensions, <math>P, Q, \operatorname{d}P, \operatorname{d}Q,</math> but it is possible to project a suggestion of what the differential features <math>\operatorname{d}p</math> and <math>\operatorname{d}q</math> are about on the 2-dimensional base space <math>X = P \times Q</math> by drawing arrows that cross the boundaries of the basic circles in the venn diagram for <math>X\!,</math> reading an arrow as <math>\operatorname{d}p</math> if it crosses the boundary between <math>p\!</math> and <math>\texttt{(} p \texttt{)}</math> in either direction and reading an arrow as <math>\operatorname{d}q</math> if it crosses the boundary between <math>q\!</math> and <math>\texttt{(} q \texttt{)}</math> in either direction. | | Drawing a venn diagram for the differential extension <math>\operatorname{E}X = X \times \operatorname{d}X</math> requires four logical dimensions, <math>P, Q, \operatorname{d}P, \operatorname{d}Q,</math> but it is possible to project a suggestion of what the differential features <math>\operatorname{d}p</math> and <math>\operatorname{d}q</math> are about on the 2-dimensional base space <math>X = P \times Q</math> by drawing arrows that cross the boundaries of the basic circles in the venn diagram for <math>X\!,</math> reading an arrow as <math>\operatorname{d}p</math> if it crosses the boundary between <math>p\!</math> and <math>\texttt{(} p \texttt{)}</math> in either direction and reading an arrow as <math>\operatorname{d}q</math> if it crosses the boundary between <math>q\!</math> and <math>\texttt{(} q \texttt{)}</math> in either direction. |
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| (p)~q~ | | (p)~q~ |
| \\[4pt] | | \\[4pt] |
− | (p)~~~ | + | (p)[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) |
| \\[4pt] | | \\[4pt] |
| ~p~(q) | | ~p~(q) |
| \\[4pt] | | \\[4pt] |
− | ~~~(q)
| + | [[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])(q) |
| \\[4pt] | | \\[4pt] |
| (p,~q) | | (p,~q) |
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| ((p,~q)) | | ((p,~q)) |
| \\[4pt] | | \\[4pt] |
− | ~~~~~q~~
| + | 17:54, 5 December 2014 (UTC)q~~ |
| \\[4pt] | | \\[4pt] |
| ~(p~(q)) | | ~(p~(q)) |
| \\[4pt] | | \\[4pt] |
− | ~~p~~~~~ | + | ~~p17:54, 5 December 2014 (UTC) |
| \\[4pt] | | \\[4pt] |
| ((p)~q)~ | | ((p)~q)~ |
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| |} | | |} |
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− | For example, given the set <math>X = \{ a, b, c \},\!</math> suppose that we have the 2-adic relative term <math>\mathit{m} = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~~~~}\, {}^{\prime\prime}</math> and | + | For example, given the set <math>X = \{ a, b, c \},\!</math> suppose that we have the 2-adic relative term <math>\mathit{m} = {}^{\backprime\backprime}\, \text{marker for}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 17:54, 5 December 2014 (UTC)}\, {}^{\prime\prime}</math> and |
| the associated 2-adic relation <math>M \subseteq X \times X,</math> the general pattern of whose common structure is represented by the following matrix: | | the associated 2-adic relation <math>M \subseteq X \times X,</math> the general pattern of whose common structure is represented by the following matrix: |
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| |} | | |} |
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− | Recognizing that <math>a\!:\!a + b\!:\!b + c\!:\!c</math> is the identity transformation otherwise known as <math>\mathit{1},\!</math> the 2-adic relative term <math>m = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~~~~}\, {}^{\prime\prime}</math> can be parsed as an element <math>\mathit{1} + a\!:\!b + b\!:\!c + c\!:\!a</math> of the so-called ''group ring'', all of which makes this element just a special sort of linear transformation. | + | Recognizing that <math>a\!:\!a + b\!:\!b + c\!:\!c</math> is the identity transformation otherwise known as <math>\mathit{1},\!</math> the 2-adic relative term <math>m = {}^{\backprime\backprime}\, \text{marker for}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 17:54, 5 December 2014 (UTC)}\, {}^{\prime\prime}</math> can be parsed as an element <math>\mathit{1} + a\!:\!b + b\!:\!c + c\!:\!a</math> of the so-called ''group ring'', all of which makes this element just a special sort of linear transformation. |
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| Up to this point, we are still reading the elementary relatives of the form <math>i\!:\!j</math> in the way that Peirce read them in logical contexts: <math>i\!</math> is the relate, <math>j\!</math> is the correlate, and in our current example <math>i\!:\!j,</math> or more exactly, <math>m_{ij} = 1,\!</math> is taken to say that <math>i\!</math> is a marker for <math>j.\!</math> This is the mode of reading that we call "multiplying on the left". | | Up to this point, we are still reading the elementary relatives of the form <math>i\!:\!j</math> in the way that Peirce read them in logical contexts: <math>i\!</math> is the relate, <math>j\!</math> is the correlate, and in our current example <math>i\!:\!j,</math> or more exactly, <math>m_{ij} = 1,\!</math> is taken to say that <math>i\!</math> is a marker for <math>j.\!</math> This is the mode of reading that we call "multiplying on the left". |
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| To construct the regular representations of <math>S_3,\!</math> we begin with the data of its operation table: | | To construct the regular representations of <math>S_3,\!</math> we begin with the data of its operation table: |
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− | {| align="center" cellpadding="6" width="90%" | + | {| align="center" cellpadding="10" style="text-align:center" |
− | | align="center" |
| + | | <math>\text{Symmetric Group}~ S_3</math> |
− | <pre> | + | |- |
− | Symmetric Group S_3 | + | | [[Image:Symmetric Group S(3).jpg|500px]] |
− | o-------------------------------------------------o
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− | | |
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− | | ^ |
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− | | e / \ e |
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− | | / \ | | |
− | | / e \ | | |
− | | f / \ / \ f |
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− | | / \ / \ |
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− | | / f \ f \ |
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− | | g / \ / \ / \ g |
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− | | / \ / \ / \ |
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− | | / g \ g \ g \ |
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− | | h / \ / \ / \ / \ h |
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− | | / \ / \ / \ / \ |
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− | | / h \ e \ e \ h \ |
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− | | i / \ / \ / \ / \ / \ i |
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− | | / \ / \ / \ / \ / \ |
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− | | / i \ i \ f \ j \ i \ |
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− | | j / \ / \ / \ / \ / \ / \ j |
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− | | / \ / \ / \ / \ / \ / \ |
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− | | ( j \ j \ j \ i \ h \ j ) |
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− | | \ / \ / \ / \ / \ / \ / |
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− | | \ / \ / \ / \ / \ / \ / |
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− | | \ h \ h \ e \ j \ i / |
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− | | \ / \ / \ / \ / \ / |
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− | | \ / \ / \ / \ / \ / |
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− | | \ i \ g \ f \ h / |
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− | | \ / \ / \ / \ / |
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− | | \ / \ / \ / \ / |
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− | | \ f \ e \ g / |
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− | | \ / \ / \ / |
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− | | \ / \ / \ / |
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− | | \ g \ f / |
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− | | \ / \ / |
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− | | \ / \ / |
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− | | \ e / |
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− | | \ / |
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− | | \ / |
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− | | v |
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− | | |
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− | o-------------------------------------------------o
| |
− | </pre>
| |
| |} | | |} |
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| Since we have a function of the type <math>L : G \times G \to G,</math> we can define a couple of substitution operators: | | Since we have a function of the type <math>L : G \times G \to G,</math> we can define a couple of substitution operators: |
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− | {| align="center" cellpadding="6" width="90%" | + | {| align="center" cellpadding="10" width="90%" |
| | valign="top" | 1. | | | valign="top" | 1. |
| | <math>\operatorname{Sub}(x, (\underline{~~}, y))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(\underline{~~}, y),</math> with the effect of producing the saturated rheme <math>(x, y)\!</math> that evaluates to <math>xy.\!</math> | | | <math>\operatorname{Sub}(x, (\underline{~~}, y))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(\underline{~~}, y),</math> with the effect of producing the saturated rheme <math>(x, y)\!</math> that evaluates to <math>xy.\!</math> |
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| In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)</math> into <math>xy,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : xy) ~|~ y \in G \}.</math> The pairs <math>(y : xy)\!</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the right margin. This produces the ''regular ante-representation'' of <math>S_3,\!</math> like so: | | In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)</math> into <math>xy,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : xy) ~|~ y \in G \}.</math> The pairs <math>(y : xy)\!</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the right margin. This produces the ''regular ante-representation'' of <math>S_3,\!</math> like so: |
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− | {| align="center" cellpadding="6" width="90%" | + | {| align="center" cellpadding="10" style="text-align:center" |
− | | align="center" |
| + | | |
| <math>\begin{array}{*{13}{c}} | | <math>\begin{array}{*{13}{c}} |
| \operatorname{e} | | \operatorname{e} |
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| In (2) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(y, \underline{~~}),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(y, \underline{~~})</math> into <math>yx,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : yx) ~|~ y \in G \}.</math> The pairs <math>(y : yx)\!</math> can be found by picking an <math>x\!</math> on the right margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the left margin. This produces the ''regular post-representation'' of <math>S_3,\!</math> like so: | | In (2) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(y, \underline{~~}),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(y, \underline{~~})</math> into <math>yx,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : yx) ~|~ y \in G \}.</math> The pairs <math>(y : yx)\!</math> can be found by picking an <math>x\!</math> on the right margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the left margin. This produces the ''regular post-representation'' of <math>S_3,\!</math> like so: |
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− | {| align="center" cellpadding="6" width="90%" | + | {| align="center" cellpadding="10" style="text-align:center" |
− | | align="center" |
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| <math>\begin{array}{*{13}{c}} | | <math>\begin{array}{*{13}{c}} |
| \operatorname{e} | | \operatorname{e} |
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| | / \ | | | | / \ | |
| | / \ | | | | / \ | |
− | | o o | | + | | o G o | |
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− | | | G | | | + | | | o<---------T---------o | |
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| | \ / | | | | \ / | |
| | \ / | | | | \ / | |
− | | \ T / | | + | | \ / | |
− | | \ o<------------/-------------o | | + | | \ / | |
| | \ / | | | | \ / | |
| | \ / | | | | \ / | |
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| # http://forum.wolframscience.com/showthread.php?postid=1602#post1602 | | # http://forum.wolframscience.com/showthread.php?postid=1602#post1602 |
| # http://forum.wolframscience.com/showthread.php?postid=1603#post1603 | | # http://forum.wolframscience.com/showthread.php?postid=1603#post1603 |
| + | |
| + | [[Category:Artificial Intelligence]] |
| + | [[Category:Boolean Algebra]] |
| + | [[Category:Boolean Functions]] |
| + | [[Category:Charles Sanders Peirce]] |
| + | [[Category:Combinatorics]] |
| + | [[Category:Computational Complexity]] |
| + | [[Category:Computer Science]] |
| + | [[Category:Cybernetics]] |
| + | [[Category:Differential Logic]] |
| + | [[Category:Equational Reasoning]] |
| + | [[Category:Formal Languages]] |
| + | [[Category:Formal Systems]] |
| + | [[Category:Graph Theory]] |
| + | [[Category:Inquiry]] |
| + | [[Category:Inquiry Driven Systems]] |
| + | [[Category:Knowledge Representation]] |
| + | [[Category:Logic]] |
| + | [[Category:Logical Graphs]] |
| + | [[Category:Mathematics]] |
| + | [[Category:Philosophy]] |
| + | [[Category:Propositional Calculus]] |
| + | [[Category:Semiotics]] |
| + | [[Category:Visualization]] |